lecture04

lecture04 - Yinyu Ye MS&E Stanford...

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Unformatted text preview: Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #04 1 TheSimplexMethodI YinyuYe DepartmentofManagementScienceandEngineering StanfordUniversity Stanford,CA94305,U.S.A. http://www.stanford.edu/yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #04 2 History GeorgeB.Dantzig s SimplexMethod forlinearprogrammingstandsasoneofthe mostsignificantalgorithmicachievementsofthe20thcentury.Itisnowover50 yearsoldandstillgoingstrong. Thebasicideaofthesimplexmethodtoconfinethesearchto cornerpoints ofthe feasibleregion(ofwhichthereareonly finitely many)inamostintelligentway.In contrast, interior-pointmethods willmoveintheinteriorofthefeasibleregion, hopingtoby-passmany cornerpoints ontheboundaryoftheregion. Thekeyforthesimplexmethodistomakecomputers see cornerpoints;andthe keyforinterior-pointmethodsisto stay intheinteriorofthefeasibleregion. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #04 3 RecallFactsofLinearProgramming AllLPproblemsfallintooneofthreeclasses: Problemis infeasible :Feasibleregionisempty. Problemis unbounded :Feasibleregionisunboundedtowardstheoptimizing direction. Problemis feasibleandbounded .Inthiscase: thereexistsan optimalsolutionoroptimizer . Theremaybea unique optimizeror multiple optimizers. Alloptimizersareona face ofthefeasibleregion. Thereisalwaysatleastone corner(extreme) optimizerifthefacehasa corner. Ifacornerpointisnot worse thanallits adjacentorneighboring corners, thenitisoptimal. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #04 4 LocalOptimalityimpliesGlobalOptimalityinConvexOptimization (P) minimize f ( x ) subjectto x R n , isa ConvexOptimization problemif f ( x ) isa convexfunction ina convex feasibleset . Proofbycontradiction .Suppose x isa localminimizer butnota globalminimizer x * ,thatis, x and x * but f ( x * ) <f ( x ) .Nowtheconvex combinationpoint x +(1- ) x * mustbefeasible(?),and f ( x +(1- ) x * ) f ( x )+(1- ) f ( x * ) <f ( x ) forany < 1 .Thiscontradictsthelocaloptimalityas canbearbitrarily closeto 1 sothat x +(1- ) x * canbearbitrarilycloseto x . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #04 5 Fromgeometrytoalgebra Howtomakecomputerrecognizea cornerpoint ? Howtomakecomputertellthattwocornersare neighboring ? Howtomakecomputer terminate anddeclareoptimality? Howtomakecomputeridentifyabetter neighboringcorner ? Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #04 6 BasicandBasicFeasibleSolution IntheLPstandardform,select m linearlyindependentcolumns ,denotedbythe variableindexset B ,from A .Solve A B x B = b forthe m-vector x B .Bysettingthevariables, x N ,of x correspondingtothe remainingcolumnsof A equaltozero,weobtainasolution x suchthat A x = b . Then, x issaidtobea basicsolution to(LP)withrespecttothe basicvariableset B .Thevariablesin x B arecalled basicvariables ,thosein x N are nonbasic variables ,and A B iscalled basis . Ifabasicsolution x B ,then x iscalleda basicfeasiblesolution,orBFS . Yinyu Ye, MS&E, Stanford...
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This note was uploaded on 05/06/2010 for the course MSE 211 at Stanford.

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lecture04 - Yinyu Ye MS&E Stanford...

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